Objections to Hilbert’s Hotel

by Luke Muehlhauser on July 9, 2009 in Kalam Argument

Part 6 of my Mapping the Kalam series.

infinityLast time, I showed how Craig & Sinclair use the thought experiment of Hilbert’s Hotel to support the proposition, “an actual infinite cannot exist.” (You’ll have to read the other posts in this series for this one to make sense.)

Today, we look at three objections to Craig & Sinclair’s use of Hilbert’s Hotel and see how the authors respond.

Craig & Sinclair claim:

Hilbert’s Hotel is absurd. But if an actual infinite were metaphysically possible, then such a hotel would be metaphysically possible. It follows that the real existence of an actual infinite is not metaphysically possible.

Oppy’s response

Graham Oppy’s response to this is to bite the bullet:

…these allegedly absurd situations are just what one ought to expect if  there were… physical  infinities. [The fan of actual infinites can] embrace the conclusion of [his] opponent’s reductio ad absurdum argument. [For example:] “They thought they had me, but I outsmarted them. I agreed that it is sometimes just to hang an innocent man.”1

Oppy’s response is to accept the absurdities of Hilbert’s Hotel, much as we accept the absurdities of quantum mechanics. The difference may be that we have tons of strong evidence for the absurdities of quantum mechanics, and perhaps we do not have good evidence for the absurdities of the actual infinite. So, say Craig & Sinclair, Oppy’s acceptance of the absurdities of Hilbert’s Hotel does nothing to show us why we should accept them despite their absurdity.

Sobel’s response

Sobel observes2 that Hilbert’s Hotel brings two seemingly irrefutable propositions into conflict:

(i) There are not more things in a multitude M than there are in a multitude M′ if there is a one-to-one correspondence of their members.

(ii) There are more things in M than there are in M′ if M′ is a proper submultitude of M.

That is common sense. But as Hilbert’s Hotel shows, (i) and (ii) are in conflict if:

(iii) An infinite multitude exists.

The obvious solution, say Craig & Sinclair, is to reject (iii), which seems much less certain than (i) and (ii). But Sobel’s choice, rather surprisingly, is to retain (iii) and reject (ii). But Sobel offers no argument for rejecting the seemingly irrefutable (ii).

Sobel also says that the absurdities of Hilbert’s Hotel “bring out the physical impossibility of this particular infinity of concurrent real things, not its logical impossibility.” But Craig & Sinclair admit the logical possibility of Hilbert’s Hotel, they simply deny its metaphysical possibility. So, Sobel’s rejection of (ii) is less plausible than Craig & Sinclair’s rejection of (iii), and Sobel’s defense of the logical possibility of Hilbert’s Hotel misses its mark.

Yandell’s response

In response to Hilbert’s Hotel, Yandell claims that subtraction of infinite quantities does not yield contradictions:

Subtracting the even positive integers from the set of positive integers leaves an infinite set, the odd positive integers. Subtracting all of the positive integers greater than 40 from the set of positive integers leaves a finite (forty-membered) set. Subtracting all of the positive integers from the set of positive integers leaves one with the null set. But none of these subtractions could possibly lead to any other conclusion than each leads to. This alleged contradictory feature of the infi  nite seems not to generate any actual contradictions.3

To which Craig & Sinclair respond:

It is, of course, true that every time one subtracts all the even numbers from all the natural numbers, one gets all the odd numbers, which are infinite in quantity. But that is not where the contradiction is alleged to lie. Rather the contradiction lies in the fact that one can subtract equal quantities from equal quantities and arrive at different answers. For example, if we subtract all the even numbers from all the natural numbers, we get an infinity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the  identical number of numbers and yet did not arrive at an identical result. In fact, one can subtract equal quantities from equal quantities and get any quantity between zero and infinity as the remainder.

So, neither Oppy nor Sobel nor Yandell can avoid the absurdities of Hilbert’s Hotel, which seem to demonstrate that “an actual infinite cannot exist,” which is in turn part of a syllogism defending the KCA’s first premise, “The universe began to exist.”

But if we can find concrete examples of actual infinities in the universe, this would defeat premise 2.11 – “An actual infinite cannot exist” – despite the strength of the analogy of Hilbert’s Hotel. We turn to this discussion next.

  1. See page 48 of Oppy’s Philosophical Perspectives on Infinity. []
  2. Logic and Theism, page 186. See the bibliography. []
  3. Infinity and Explanation, Damnation and Empiricism in Does God Exist edited by Craig, Flew, and Wallace. []

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{ 67 comments… read them below or add one }

urbster1 July 9, 2009 at 6:27 pm

I never understood this business about “absolute infinites” not being able to exist. What about God? Isn’t God supposed to be infinitely good, and infinitely just, infinitely loving, infinitely caring, infinitely merciful, and so on? Or what about being infinitely timeless, if there is such a thing? If actual infinites cannot exist, then God’s existence should be ruled out too. Why isn’t this used as an objection to the argument?

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Kevin July 9, 2009 at 6:58 pm

urbster1,
Craig has this covered.  When the theist says God is infinitely powerful, knowing, etc., he doesn’t mean that knowledge or power come in discrete units, and God has an infinite amount of these respective units.  Rather, infinite power, love, goodness, knowledge, and so on (or better, omnipotence, omnibenevolence, omniscience, and so on) means that God can do or be anything it is possible to do or be with respect to each of these.
Thus, saying God is omnipotent doesn’t mean he has infinitely many units of power, but that he can do anything it is possible to do.  Being infinitely good means everything he wills is good, and nothing he wills is evil.  And so on.
I suppose with respect to omnipotence this might mean that, numerically, there are a finite number of actions God can perform. But this is so only if the number of logically possible actions is limited, and that doesn’t seem absurd on the face of it.
As for my own comments on Hilbert’s Hotel, I’ve always found it absurd that a new visitor could show up to the Hotel and request a room. If there are an infinite number of guests already, then wouldn’t this exhaust all possible guests?  Where can a new guest come from, if all possible guests are already in the hotel?  If a new guest shows up, then the hotel does not have an infinity of guests already.  Analogously, how can one add any quantity to infinity, if infinity already contains all numbers.
I’ve no doubt there’s a resonable response to these points from Set Theory, so if anyone can clue me in, please do.

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Lorkas July 9, 2009 at 7:37 pm

Kevin: I suppose with respect to omnipotence this might mean that, numerically, there are a finite number of actions God can perform. But this is so only if the number of logically possible actions is limited, and that doesn’t seem absurd on the face of it.

I’m confused about what you’re saying here. Are you arguing that God’s power is finite or infinite?

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Hylomorphic July 9, 2009 at 8:14 pm

Oppy’s response strikes me as the most likely to pan out. Craig’s objection to the possibility of an actual Hilbert’s Hotel seems to me point out nothing more than that our intuitions for dealing with finitude don’t work very well for infinitely large things.
 
I’m hardly a mathematician, but I fail to see why would this be at all shocking or surprising. Why should this failure of intuition be taken as a reason to think an actual infinite impossible?
 
Oppy need not give us a reason to think that actual infinites are possible. Without a good reason to reject the notion, we ought to accept that possibility, just as we do with other notions. Oppy need only point out in what way Craig fails to give us a reason to reject the possibility.
 
Bill Craig wrote:

For example, if we subtract all the even numbers from all the natural numbers, we get an infi  nity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the  identical number of numbers and yet did not arrive at an identical result.

 
Oh, really? You subtracted the identical number of numbers in both cases? Exactly what number of numbers was that?

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Taranu July 9, 2009 at 10:17 pm

It seems to me that if the B theory of time is correct, whenever we are talking about an infinity of temporal events we cannot subtract anything because the past, present and future are fixed. In this case Hilbert’s Hotel is no longer a valid analogy. It relies on subtraction being possible.

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Matthew D. Johnston July 9, 2009 at 10:41 pm

This argument always seemed strange to me – I’m a mathematician, but not a set theorist, so I have some background in this but it’s not a matter of intimate concern. Hilbert’s Hotel is typically given as a second-year (or advanced first-year) introductory example of the difficulty of conceptualizing infinite sets. Craig omits the next lecture where we learn how to use the concept anyway.
In short, I strongly doubt any useful statement about the existence of actual infinities can come from such an exercise, whether they’re true or not.
Consider this: Given any two points in space, how many points lie on the line segment connecting them?
The answer is (probably) infinite, but the objection might be raised that this is a different kind of infinite than Craig is interested it (it is uncountable), so we will modify out approach.
Call one point 0 and the other point 1 and consider the set of points n/m, where m > n (n and m are positive integers).
This is an infinite set of points lying between 0 and 1, but not only that, it shares a one-to-one correspondence with the positive integers, which are countable (this can be proved mathematically).
In other words, these points can be indexed exactly like the rooms of Hilbert’s Hotel. Consequently, all paradoxes of infinity apply equally to the Hotel as they do the collection of points in space. If the paradox really implies that actual infinities cannot exist, this would seem to imply that space (and time) are necessarily non-continuous – rather, they are divided into discrete points. I do not feel anybody would be willing to make such a conclusion based on a thought experiment rather than hard data.

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lukeprog July 9, 2009 at 10:51 pm

Matthew,

Is there an online lecture or book chapter you could point me to regarding that “second lecture” wherein “we learn how to use the concept anyway”?

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Steven Carr July 9, 2009 at 11:07 pm

CRAIG
For example, if we subtract all the even numbers from all the natural numbers, we get an infinity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers.

CARR
I guess Craig flunked maths.

Is Craig now claiming that the number of positive integers is NOT infinite?

HYLOMORPHIC
You subtracted the identical number of numbers in both cases? Exactly what number of numbers was that?

CARR
Brilliant. Craig has no idea how to handle infinite series.

Not only does Craig’s intuition tell him infallibly what sort of things must happen when the entire universe is smaller than a proton, but his intuition is more reliable about infinite series than the combined brains of the best mathematicians who have worked on the subject.

There are amusing jokes in mathematics, where people take the series 1,2,4,6,8,16,32,64,128, etc and add up the terms using standard maths to get the answer , which turns out to be minus 1.

This is a nice mathematical joke, but Craig uses similar techniques in all seriousness as part of his philosophy.

Perhaps Craig has now ‘proved’ that we cannot multiply things by 2?

For proof of the sum being -1.

Let S = 1 + 2 + 4 + 8 + 16 + 32 + 64 etc

Double everything,
2S = 2 + 4 + 8 + 16 + 32 + 64….

Subtract 1 equation from another
S – 2S = 1

Therefore S = -1 QED

Most mathematicians just have a wry smile at such jokes, but Craig puts these idea of taking one infinity from another in his philosophical papers.

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Taranu July 9, 2009 at 11:39 pm

Steven, could you point out a link where such math jokes can be found? I would like to know more about them.

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Ryan July 10, 2009 at 12:59 am

This is an excellent post! I wish more atheist blogs were like this!

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Rick July 10, 2009 at 3:02 am

One of the first things I did when I found out about infinity is to treat it like a number. Even my algebra teacher knew that to do so was a grave mathematical error, but it seems that Craig wishes to hinge a major objection in his argument on his misunderstanding of maths. First of all, semantics is very important:
 

For example, if we subtract all the even numbers from all the natural numbers, we get an infinity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the  identical number of numbers and yet did not arrive at an identical result.
 

I’m fairly sure that he meant, ‘if we subtract the set of natural numbers greater than three…’ Yes, I’m nitpicking, but if he’s going to hinge his rejection of infinity on this, he needs to be particular. However, \infty \neq \infty to put it in \LaTeX form. Just because one set is infinite does not mean it’s as large as the next infinite set. So in comparing the set of even numbers to the set of numbers greater than three, he misuses the concept of infinity as though it were an actual quantity, making the same mistake I was corrected for in junior high.
 
IMO, it damages his credibility if he’s to argue about Hilbert’s Hotel or the mathematics of infinity and insists on making these kinds of mistakes.

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Reginald Selkirk July 10, 2009 at 5:49 am

I brought this up in the earlier thread, but this looks like a reasonable place to repeat. My response is in agreement with Stephen Carr.Craig does not understand the mathematics of infinity.

To which Craig & Sinclair respond:

It is, of course, true that every time one subtracts all the even numbers from all the natural numbers, one gets all the odd numbers, which are infinite in quantity. But that is not where the contradiction is alleged to lie. Rather the contradiction lies in the fact that one can subtract equal quantities from equal quantities and arrive at different answers. For example, if we subtract all the even numbers from all the natural numbers…

This is the bit where Craig claims to have derived 0 = 3 by subtracting one infinity from another. It seems to me that this objection would apply not just to actual physical infinities, but to theoretical mathematical infinities as well. I.e. the entire mathematics of infinity would have been wiped out reductio ad absurdum if Craig’s use of math was correct.
Other objections to Hilbert’s Hotel mentioned previously: Note that no difficulties show up due merely to having a hotel with an infinite number of rooms. They only show up when one considers a separate infinity, an infinite number of guests, and then plays off one infinity against the other.
Another objection previously raised: that any difficulties with infinities of matter and space do not necessarily translate to a timeline with an infinite past.
 

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lukeprog July 10, 2009 at 7:13 am

Thanks, Ryan!

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corn July 10, 2009 at 7:15 am

It seems to me that the entirety of Craig’s argument could be dispatched having taken only three undergraduate courses: Epistemology, Number Theory, and Comparative Religion. Unfortunately most people don’t believe things because of logical arguments for and against a particular notion. I don’t take the KCA seriously because it’s not reasoning that led to a conclusion, but reasoning derived from the conclusion. For those that believe in the conclusion, the reasoning is sound. For those that don’t, the reasoning is often easily disproved. Sometimes, as with the case of countably infinite sets, uncountably infinite sets, and different magnitudes of infinity, you need particular background to understand why a premise is incorrect. Take the sum of integers n where n = 2(n-1). John Carr used simple algebra to show that this sum is -1. However no one questioned the assumption that |S| > |2S| which is what allows John to arrive at his sum. That is, the number of members of S is exactly one greater than the number of members of 2S. This leads his setup of the problem to look like:

 S = 1 + 2 + 4 + 8 + 16 + 32 + ...
2S =     2 + 4 + 8 + 16 + 32 + ...
S - 2S = (1-∅) + (2-2) + (4-4) + (8-8) + ...

when in fact it should be setup as:

S = 1 + 2 + 4 + 8 + 16 + 32 + ...
2S = 2 + 4 + 8 + 16 + 32 + 64 + ...
S - 2S = (1-2) + (2-4) + (4-8) + (8-16) + ...
-S = -1 + -2 + -4 + -8 + ...

In this case we have simply arrived at the original problem of an infinite sum being the difference between two infinite sums. That is to say, we can not apply traditional algebraic sums to all infinite quantities. Craig would do well to study more Georg Cantor before postulating on set arithmetic.

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lukeprog July 10, 2009 at 7:21 am

corn,

I could use some number theory. Any modern summary books you’d recommend that would be helpful to this end?

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Dave July 10, 2009 at 7:59 am

Luke, I’m afraid both you and Craig have mischaracterized Sobel’s position. Sobel does not hold that (ii) should be rejected, only that it should be restricted to apply only to finite multitudes. Were he to explicitly spell out his argument — and having cited “a preponderance of the mathematical community,” perhaps he felt he didn’t need to — then I imagine it would go like this:

(iia) For finite multitudes M and M’, there are more things in M than there are in M′ if M′ is a proper submultitude of M.
(iib) For finite or infinite multitudes M and M’, there are more things in M than there are in M′ if M′ is a proper submultitude of M.

There is no good reason to prefer (iib) over (iia), and there is no contradiction between (i), (iia) and (iii). Thus, it seems to be Craig, not Sobel, who has the burden of proof here.

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Gordon Reid July 10, 2009 at 8:00 am

The Craig & Sinclair argument leads to the proof that the actual universe is not metaphysically possible.
 

Assume Craig & Sinclair that an actual infinite is not metaphysically possible.
Choose any two points in the universe.
The straight line between these two points is made up of an infinite number of points.
But according to 1, these actual points are not metaphysically possible and by extension all points in the universe are not metaphysically possible.
Therefore, the universe is not metaphysically possible.

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Lorkas July 10, 2009 at 8:06 am

Gordon Reid: The straight line between these two points is made up of an infinite number of points.

Unless space isn’t continuous.

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Kevin July 10, 2009 at 8:07 am

Lorkas,
I was arguing that “infinite” is perhaps not the proper term to apply to God’s power, knowledge, etc., since it implies quantity.  Craig and other apologists reject the notion that God’s power is a matter of quantity.  Instead, saying God is all-powerful means that he can do anything it is (logically) possible to do, and which doesn’t conflict with other aspects of his nature.  Given this, the number of unique actions God can perform might be finite, but only because the number of logically possible actions is finite.
The number of possible actions for God might be further limited by potential conflicts with other aspects of his nature.  Thus, there may be many logically possible actions that are evil, and so God would not be able to perform them.

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Gordon Reid July 10, 2009 at 8:20 am

Lorkus:
Agreed…but if space is not continuous, then Craig’s Hilbert’s Hotel does not have an infinite number of rooms.

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Lorkas July 10, 2009 at 8:59 am

Kevin: Instead, saying God is all-powerful means that he can do anything it is (logically) possible to do, and which doesn’t conflict with other aspects of his nature. Given this, the number of unique actions God can perform might be finite, but only because the number of logically possible actions is finite.

Sure, it might be finite, but I don’t see any reason to suppose it is. In any case, omnipotence isn’t just about the number of actions you can perform–it also has to do with the degree to which you can do an action, like apply a force. There isn’t a limit to the amount of force that can be applied, so you have to accept that God has to be infinite in this respect, or accept that he’s finite in that respect.
 
If he’s finite and not maximal, then it is possible that a being could be more powerful than God, an idea that many theists are uncomfortable with. If he’s infinite in this respect, then the premise that an actual infinite can’t exist is false, which introduces a contradiction into the Kalam, rendering it invalid.

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Lorkas July 10, 2009 at 9:00 am

Gordon Reid: Lorkus: Agreed…but if space is not continuous, then Craig’s Hilbert’s Hotel does not have an infinite number of rooms.

That doesn’t follow at all. It’s possible to conceive of a space that’s continuous but infinite in dimension. It would be rather like the set of all integers: it is divided into discrete units, but is infinite.

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Lorkas July 10, 2009 at 9:01 am

Lorkas: It’s possible to conceive of a space that’s continuous but infinite in dimension.

Oops! I mean, “discontinuous but infinite in dimension”. I think that this can be understood from my analogy.

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Lamplighter Jones July 10, 2009 at 9:33 am

Two remarks:
1.  The assertion that actual infinities contradict (ii), and therefore there are no actual infinities, is question-begging.
(ii) says There are more things in M than there are in M′ if M′ is a proper submultitude of M.
To be precise, (ii) says that If M’ is a proper submultitude of M (that is, every member of M’ is a member of M, and there is a member of M which is not a member of M’), then M’ and M are not in bijection.
But the statement “M is infinite” is an abbreviation of the statement “M has a proper submultitude M’ which is in bijection with M,” so the argument “actual infinities violate (ii), so there are no actual infinities,” is an abbreviation of the argument “(ii) is true, therefore (ii) is true.”
 
2. Criag’s objection to the existence of actual infinities is a special case of the following: there is a property of finite collections that is not shared by infinite collections.  Therefore infinite collections cannot exist.
Really, it appears that Craig is objecting to the idea that two distinct objects can have different properties.  He does nothing to explain what is special about properties (i) and (ii).

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Gordon Reid July 10, 2009 at 9:38 am

Lorkas:
Technically you are correct but this is the very point where I think Craig’s analogy fails.  Any limited set of integers is finite and the math of addition, subtraction, etc., works.   The set of all integers is infinite and addition, subtraction, etc., becomes indeterminate.  Corn above explains this perfectly.  In my comment about Craig’s Hilbert’s Hotel I assumed the set of discrete points in a line in a discontinuous universe could be drawn from one end of the hotel to the other and thus the hotel would have a finite length just as the set of point in this length is finite even though the universe is of infinite dimensions.

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Kevin July 10, 2009 at 9:47 am

Lorkas: omnipotence isn’t just about the number of actions you can perform–it also has to do with the degree to which you can do an action, like apply a force. There isn’t a limit to the amount of force that can be applied, so you have to accept that God has to be infinite in this respect,

Good point.  I’m sure Craig would have a response to this, though.  I don’t agree with him, but I don’t underestimate him either.
There might be a limit to the amount of force God can apply if there is a limit determined by the laws of nature (sort of like light speed being a limit to velocity). But then, God determines these laws of nature, and perhaps he could have done things otherwise, such that there would be no limit to force (or to velocity).
But even if this is the case, this might only mean that God’s ability to apply force is potentially infinite.  Could he ever apply infinite force?  Perhaps not if he’s starting from zero.
But this also occurs to me (sorry, this is sort of stream-of-consciousness): Perhaps the notion of applying infinite force is contradictory.  Force is applied to something that resists it.  Such resistence is typically finite, so any force needed to overcome it would be finite too.  But if such resistence is infinite, then no amount of force would overcome it–the best you could do with infinite force is reach a balance or stalemate.
On the other hand, would not such an infinite resistence be the equivalent to an infinite force?  If so, can we have two infinite applications of force?
I’m rambling now, so I’ll stop.  It seems like I’ve invented a convulated version of the question, What happens when an irresistable force meets an immovable object.

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Reginald Selkirk July 10, 2009 at 11:18 am

Kevin: But this also occurs to me (sorry, this is sort of stream-of-consciousness): Perhaps the notion of applying infinite force is contradictory. Force is applied to something that resists it. Such resistence is typically finite, so any force needed to overcome it would be finite too. But if such resistence is infinite, then no amount of force would overcome it–the best you could do with infinite force is reach a balance or stalemate.

It appears you just re-invented the old argument, “Could God create a rock so large that God could not move it?”
Rather than introducing new ambiguities about an object “resisting” force, why not use the well-established formulae, such F = MA ?
 

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Paul July 10, 2009 at 12:36 pm

Could one consider PI as an actual infinite?

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Lorkas July 10, 2009 at 4:20 pm

Kevin: There might be a limit to the amount of force God can apply if there is a limit determined by the laws of nature (sort of like light speed being a limit to velocity)

God isn’t limited by the laws of nature.

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lukeprog July 10, 2009 at 6:14 pm

Dave,

Thanks, I’ll return to this later.

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Dace July 11, 2009 at 1:05 am

A couple points that I haven’t seen raised yet:
1 – Can Craig and Sinclair justify the separation of logical and metaphysical possiblity on which their argument relies? (Does anyone here have an argument to that effect?)
2 – If an actual infinity is logically possible, then what in heck are Craig and Sinclair doing talking about where the contradiction lies anyway? Absurdity perhaps; contradiction. no. It’s like they threw in the qualifier concerning logical versus metaphysical possibility after the fact, realizing that their argument was bunk without it.

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lukeprog July 11, 2009 at 5:55 am

Dace,

Those two issues are new to me, too. I’m used to a distinction between logical and physical possibility, but I’m not sure how one would prove/disprove metaphysical possibility. And yeah – contradiction doesn’t seem like the right word for metaphysical impossibility.

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Kevin July 11, 2009 at 8:39 am

Lorkas,
Yes, God isn’t limited by the laws of nature, and I think I meant to cover that, but must have forgotten.  To change the subject a bit, apologists seem to forget this fact when it comes to the fine tuning argument.  The argument seems to assume, if it doesn’t explicitly state, that life can only exist within a narrow range of variables, circumstances, etc.  Thus, an intelligent being must have fine tuned things for life.  But surely this would be no obstacle for a God.  God should be able to create life under any natural conditions.  A better demonstration would be life existing in an environment that all other evidence says it shouldn’t.

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TK July 11, 2009 at 9:54 am

I think the real problem with applying Hilbert’s Hotel to an infinite regress of past events is that the so-called “absurdities” seem to apply to infinite sets of physical objects–things upon which we can perform operations like addition, subtraction, moving around, etc.
 
An infinite regress of past events is not an actually infinite set of physical objects. I can’t double the number of past events, hold them in my hand and move them around, add to them, or anything like that. Hilbert’s Hotel simply doesn’t apply.

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Steven Carr July 11, 2009 at 10:33 am

TK
I can’t double the number of past events, hold them in my hand and move them around, add to them, or anything like that.
 
CARR
Correct.
 
We can show that there are as many even numbers as integers by showing there is a method of constructing a 1-2-1 relationship between every member of both sets.
1-2, 2-4, 3-6, 4-8,5-10 etc
 
Craig’s objection is like asking :- ‘What if more even numbers come along? Where are you going to put them?’
 
But where are these extra even numbers going to come from? We already assigned every even number to an integer. We are not going to get any more even numbers turning up.
 
If Hilbert’s Hotel has an infinite number of rooms, and more guests come along, just build more rooms.
If Craig can produce more guests from nowhere, then we can produce more rooms from nowhere.
 

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Dace July 11, 2009 at 4:16 pm

lukeprog: Dace,Those two issues are new to me, too. I’m used to a distinction between logical and physical possibility, but I’m not sure how one would prove/disprove metaphysical possibility. And yeah – contradiction doesn’t seem like the right word for metaphysical impossibility.

I think if Craig and Sinclair consistently replaced ‘contradiction’ with ‘absurdity’ throughout, their argument would begin to look very weak: “So, the existence of an actual infinite is absurd but not contradictory. And so what? That just means it is difficult to think about, not that it couldn’t exist.”

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lukeprog July 12, 2009 at 5:39 am

exapologist has also posted Craig’s Blunder, in which he contends that Craig commits the fallacy of composition.

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corn July 12, 2009 at 7:56 am

lukeprog: corn,I could use some number theory. Any modern summary books you’d recommend that would be helpful to this end?

Number theory is quite a broad area of study. All of the books I’ve used (both when learning and teaching) are standard undergraduate text books. However what may be more useful when encountering these types of problems is a book like “Mathematical Fallacies and Paradoxes.” This book isn’t about pure mathematics per se but rather how flaws in understanding problems and applying incorrect reasoning leads to the conclusion that the sum of the positive integers is -1.

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lukeprog July 12, 2009 at 9:19 am

corn,

Thanks. That book is $0.01 used on Amazon, so I just bought it.

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Reginald Selkirk July 12, 2009 at 10:09 am

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Lamplighter Jones July 12, 2009 at 11:11 am

Apologies for my last post, which was more than a little glib.
 
I would like to point out something that’s missing from the discussion of the Kalam argument so far: an explicit definition of the terms “finite” and “infinite.”  There are several reasons why it is important to have an explicit definition of these terms.
 
1.  There are different definitions of “finite” which are not logically equivalent.  With some of these definitions, the argument in the Kalam against the existence of actual infinities really is begging the question.
 
2.  Some of the explicit definitions of “finite” make reference to numbers.  If these references entail some form of mathematical realism, then this undercuts the argument that mathematical examples of infinity are not necessarily metaphysically valid examples of infinity.  It appears that the Blackwell Companion article implicitly uses one of these definitions, since the rooms in the Hilbert’s Hotel example are all labeled with distinct integer numbers.
 
3.  To be as generous as possible to the Kalam argument, we should find a definition of “finite” that avoids the issues in 1. and 2.  (I don’t know whether or not there is such a definition.)  Furthermore, the definition of “finite” (or “infinite”) best suited to the Kalam argument should entail a “Hilbert’s Hotel”-type situation as a metaphysical possibility.
 
I’ve been too lazy to look at some of the other articles Craig has written on finite/infinite issues, and I wouldn’t be surprised if some of his writings reveal a preferred definition of “finite”.  Nevertheless, I think that the absence of an explicit definition of “finite” and “infinite” in the Blackwell Companion is a serious flaw.*
 
*- there may be an explicit definition of “finite” or “infinite” in the companion that I’ve missed. (I don’t think the discussion of “actual infinity” versus “potential infinity” counts as a definition of infinite.)

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exapologist July 12, 2009 at 12:02 pm

Hi you guys,
I should note that my post, “Craig’s Blunder”, I don’t argue that *Craig* commits the fallacy of composition. Rather, Craig attributes the fallacy to *J.L. Mackie*. And my point is that Craig is mistaken in doing so.
Best,
EA

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lukeprog July 12, 2009 at 1:42 pm

Reginald,

Yeah, I noticed the bibliography for Craig’s article only listed one of Morriston’s articles!

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lukeprog July 12, 2009 at 1:47 pm

exapologist,

Oops, that’s what I get for not reading your post before cataloging it for future reading!

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exapologist July 12, 2009 at 4:28 pm

Yeah, I noticed the bibliography for Craig’s article only listed one of Morriston’s articles!
And what’s especially frustrating to me about this is that, although he cites one of the articles in which Morriston critiques Craig’s a priori arguments against the traversability of actual infinites, he doesn’t address them. Whether Craig intended to or not, it can give the impression that Craig has dealt with the relevant criticisms from Morriston (i.e., those upon which Craig’s whole argument arguably hangs, viz., the a priori arguments against the existence and traversability of actual infinites), when in fact he hasn’t.

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exapologist July 12, 2009 at 4:29 pm

whoops — only the first sentence was supposed to be an indented quote. Sorry!

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Lamplighter Jones September 27, 2009 at 4:12 pm

J.P. Moreland’s article, A response to a Platonistic and to a set-theoretic objection to the Kalam cosmological argument, is intended to address some of Morriston’s objections to the KCA.

Having read that, I have the following question for anyone who finds the Hilbert’s Hotel scenario(s) absurd. Supposing that there is a hotel with infinitely many rooms, which of the following scenarios do you find absurd?

I. On Monday, infinitely many guests check in, filling all the rooms. On Tuesday, all of the guests check out.

II. On Monday, infinitely many guests check in, filling all the rooms. On Tuesday, all of the guests check out. On Wednesday, a new guest, different from all those who occupied the hotel on Monday night, checks in to Room 1.

III. On Monday, infinitely many guests check in, filling all the rooms. On Tuesday, all of the guests check out. On Wednesday, a new guest, different from all those who occupied the hotel on Monday night, checks in to Room 1. On Thursday, all of the guests who checked in on Monday return. The guest who occupied Room 1 on Monday occupies Room 2 on Thursday, the guest who occupied Room 2 on Monday occupies Room 3 on Thursday, and in general, the guest who occupied Room n on Monday occupies Room n+1 on Thursday.

IV. The events in scenario III all occur in the same order, but they happen over the course of four hours rather than four days.

Each of I-IV is a separate scenario.

I ask about these scenarios are because they are similar to the ones discussed by Moreland – he does not say much about the even/odd partition. (Unfortunately, a subscription is required to view Moreland’s article.)

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lukeprog September 27, 2009 at 6:42 pm

LampLighter Jones,

Check my bibliography.

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Lamplighter Jones September 29, 2009 at 12:26 pm

My bad. I should have linked here for Moreland’s paper, where it’s free.

Thanks Luke!

I would ask you to add the link to my last post, but it seems like wasted effort since the thread is so old.

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John January 10, 2010 at 4:03 pm

Kevin: urbster1,
Craig has this covered.  When the theist says God is infinitely powerful, knowing, etc., he doesn’t mean that knowledge or power come in discrete units, and God has an infinite amount of these respective units.  Rather, infinite power, love, goodness, knowledge, and so on (or better, omnipotence, omnibenevolence, omniscience, and so on) means that God can do or be anything it is possible to do or be with respect to each of these.
Thus, saying God is omnipotent doesn’t mean he has infinitely many units of power, but that he can do anything it is possible to do.  Being infinitely good means everything he wills is good, and nothing he wills is evil.  And so on.
I suppose with respect to omnipotence this might mean that, numerically, there are a finite number of actions God can perform. But this is so only if the number of logically possible actions is limited, and that doesn’t seem absurd on the face of it.
As for my own comments on Hilbert’s Hotel, I’ve always found it absurd that a new visitor could show up to the Hotel and request a room. If there are an infinite number of guests already, then wouldn’t this exhaust all possible guests?  Where can a new guest come from, if all possible guests are already in the hotel?  If a new guest shows up, then the hotel does not have an infinity of guests already.  Analogously, how can one add any quantity to infinity, if infinity already contains all numbers.
I’ve no doubt there’s a resonable response to these points from Set Theory, so if anyone can clue me in, please do.  

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John January 10, 2010 at 4:07 pm

I am a christian and I understand what you mean in regards to omnipotence, and omnibenevolence.

But what about omniscience? This seems hardest for Craig to defend.

Does God KNOW an infinite amount of things?

Let me give you an example:

Does God NOT know how many times I say hallelujah into eternity?
If I exist forever with God, then I will say “hallelujah” an infinite amount of times.

Does God have the ACTUAL conception of this infinite amount of times I say this, or does He not know?

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Timothy Underwood May 23, 2010 at 12:49 am

Someone possibly mentioned this, and I may be missing something obvious. But infinity isn’t a number. So Craig’s arguments are irrelevant.

The two values have the same cardinality, but that is the same thing as having the same number of items only for a finite cardinality.

Is anyone aware of something obvious which I am missing?

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Jim June 16, 2010 at 7:10 am

[quote]To which Craig & Sinclair respond:

It is, of course, true that every time one subtracts all the even numbers from all the natural numbers, one gets all the odd numbers, which are infinite in quantity. But that is not where the contradiction is alleged to lie. Rather the contradiction lies in the fact that one can subtract equal quantities from equal quantities and arrive at different answers. For example, if we subtract all the even numbers from all the natural numbers, we get an infi nity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the identical number of numbers and yet did not arrive at an identical result. In fact, one can subtract equal quantities from equal quantities and get any quantity between zero and infinity as the remainder.[/quote]

This response completely misses the point. Example:

Suppose that I have a stick that is 3 inches long and I remove a 1 inch long section from the stick. Should I always expect get the same result, regardless of how I choose to remove the 1 inch section? Of course not.

If I remove the 1 inch section from one end, I am left with a single 2 inch long section. If I remove the 1 inch section from the middle of the original, I am left with 2 pieces of length 1 inch each. Does this apparent “contradiction” prove that there cannot be an actual 3 inch mong stick?

Craig and Sinclair make the elementary mistake of confusing the operations of set difference and subtraction of numbers. In the world of finite sets, this creates no problems. But it just doesn’t work like that with infinite sets. There is no contradiction. There is just a mathematical mistake by C&S

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dlewisa July 28, 2010 at 5:47 pm

The hotel is an odd paradox. If we assume that there are in infinite number of occupied rooms then we must assume that there are an infinite number of people to occupy them . . . so where does this +1 person, this would-be occupier, come from? I think I need to smoke something to think about this any further . . . anyone have Willie Nelson’s or Woody Harrelson’s number?

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JOJO JACOB February 24, 2011 at 9:04 am

You do not have to be a philosopher to refute Craig’s infinity argument. The Concept of Infinity is not well-defined for Cardinal numbers. But it is for ordinal numbers. Craig says infinity minus infinity brings contradiction. What is zero minus zero? In a sense, subtracting zero from zero is logically impossible. How can you subtract nothing from nothing? Listen to Craig Vs Dr. Ahmed debate. Ahmed proved how you could confront the so-called “sophisticated” arguments using simple “common-sense.”

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John Guidone May 29, 2011 at 1:32 pm

Distinctions are being overlooked. Infinity and eternality are not the same. God is an eternal being while his nature/ attributes are infinite.

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JOJO JACOB May 30, 2011 at 12:34 am

Distinctions are being overlooked. Infinity and eternality are not the same. God is an eternal being while his nature/ attributes are infinite.

So, it is logically possible that the universe is eternal and its “lawfulness” is infinite!!!

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John Guidone May 30, 2011 at 10:53 am

Obviously not, since it BEGAN to exist. It’s lawfulness is sustained as long as the intelligent designer allows it to continue. If so, entropy will run its course on its own, or the designer will intervene with a singularity (supernatural event).

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JOJO JACOB May 31, 2011 at 12:44 am

John Guidone,

You claim that the universe began to exist. What was the state of affairs before the big bang? You might say nothing. Does your nothingness include energy? Craig says nothing means non-being. What do you mean by non-being? You can’t stretch the term “non-being” to encompass “deviations” of all those possible state of affairs in which one would naturally something/somebody a “being.” In other words, theists say that they cannot define what is non-being. But they know what is non-being. It seems to me that this is nothing more than academic word game.

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JOJO JACOB May 31, 2011 at 12:45 am

John Guidone,

You claim that the universe began to exist. What was the state of affairs before the big bang? You might say nothing. Does your nothingness include energy? Craig says nothing means non-being. What do you mean by non-being? You can’t stretch the term “non-being” to encompass “deviations” of all those possible state of affairs in which one would naturally something/somebody a “being.” In other words, theists say that they cannot define what is non-being. But they know what is not non-being. It seems to me that this is nothing more than academic word game.

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JOJO JACOB May 31, 2011 at 2:48 am

Obviously not, since it BEGAN to exist. It’s lawfulness is sustained as long as the intelligent designer allows it to continue. If so, entropy will run its course on its own, or the designer will intervene with a singularity (supernatural event).

What prevents someone from making this inference?

It’s “lawlessness” is sustained as long as the intelligent designer allows it to continue. If so, entropy will run its course on its own, or the designer will intervene with a singularity (supernatural event) to create lawfulness!!!!

You could always defend any doctrine saying might, might, might, might, might, might, might, might……………….

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John G. May 31, 2011 at 5:38 pm

If something exists now then something must exist eternally or necessarily, otherwise there would have been a period of time in the past when there was absolutely nothing. Since nothing is not a place and fields, waves and sub-atomic particles are in the universe, there was never a time that there was absolutely nothing. So the one entity or being that has always existed must have the power of being residing within itself and be a creating agent responsible for everything else that has began to come into existence. The eternally existent creating agent must also have the power and providence/sovereignty to sustain or destroy what has been created.

The notion of “chance” (which has no power or being); as being responsible for the origin of the universe is absurd since chance is just a mathematical expression to signify possibilities. Chance is nothing. Chance requires something to act upon. If you going to assign or invoke chance as the creating agent of the universe, ad hoc, you are falsely assuming that matter itself is eternal and then you must be prepared to acknowledge that chance could not sustain the universe continually. Chance implies accident, so it would stand to reason that the universe would have accidentally destroyed itself thousands of years ago. Whatever is the eternal self-existing agent transcends the universe.

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JOJO JACOB June 1, 2011 at 12:43 am

John G,

You have still not defined “non-being”!!!! On what basis do you call the state of affairs before the big bang “Nothing”. God existed before the big bang. It means there was something!!!!

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JOJO JACOB June 1, 2011 at 12:45 am

Using the same arguments/logic used by the so-called sophisticated theologians, a better case could be made for a technologically superior alien civilization who created our universe just for fun!!!!

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JOJO JACOB June 1, 2011 at 2:01 am

The notion of “chance” (which has no power or being); as being responsible for the origin of the universe is absurd since chance is just a mathematical expression to signify possibilities. Chance is nothing. Chance requires something to act upon. If you going to assign or invoke chance as the creating agent of the universe, ad hoc, you are falsely assuming that matter itself is eternal and then you must be prepared to acknowledge that chance could not sustain the universe continually. Chance implies accident, so it would stand to reason that the universe would have accidentally destroyed itself thousands of years ago. Whatever is the eternal self-existing agent transcends the universe.

John,

If chance is real , God must exist as the Cosmic Observer. If determinism is real, God exists as the Hidden Variable that stops the infinite regress of causes. You have not proved the existence of God……

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JOJO JACOB June 1, 2011 at 3:28 am

John,

“Whatever is the eternal self-existing agent transcends the universe.”

Why?

Read Paul Tillich and Dietrich Bonhoeffer.

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John G. June 1, 2011 at 9:52 am

God transcends the universe in his being AND spatially. He is boundless and exists beyond space and time and beyond our finite minds to completely comprehend. We can apprehend it , but not fully comprehend it. The problem with atheism is they cannot come to terms with the mind limitations of humanity. You cannot know everything, otherwise you would have the same attribute (all- knowing) of the being you deny exists!
There is no reason to get cranky over this. If your areconfident through your exhaustive study, that the probaility evidence confirms you have made the right choice in Pascal’s Wager and Occam’s Razor you have nothing to worry about.

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